Lorenzvergelijkingen

The Lorenz equations are a system of ordinary differential equations first studied by Edward Lorenz. They are notable for having chaotic solutions for certain parameter values and initial conditions. The equations are derived from simplified models of atmospheric convection, but they are often used to illustrate chaos theory. The Lorenz equations are:

dx/dt = σ(y - x)

dy/dt = x(ρ - z) - y

dz/dt = xy - βz

where x, y, and z are functions of time, and σ, ρ, and β are positive real parameters. The Lorenz equations exhibit chaotic behavior for certain values of the parameters, such as σ = 10, ρ = 28, and β = 8/3. In this case, the solutions are sensitive to initial conditions, meaning that small differences in the initial values of x, y, and z can lead to large differences in the solutions over time. This sensitivity to initial conditions is a key characteristic of chaotic systems. The Lorenz equations have been studied extensively in the fields of mathematics, physics, and engineering, and they continue to be an important topic of research in chaos theory.